Law of conservation of angular momentum

Angular momentum is a sort of movement caused by inertia upon the rotational body. That moment of inertia is termed the angular momentum. Here, we will discuss its conservation law, referred to as the ‘Law of conservation of Angular momentum’. In a nutshell, this Law depicts that a body or say, an object has the same (constant) angular momentum before and after exchange. Moreover, it is symbolised as the arrow heading towards the right side on the top of the upper case letter ‘L’. The topic is broad, though a complete and curated law of conservation study material is given below, which covers the Law of conservation of angular momentum formula, statement, along with an example and solves the problem. 

Law of conservation of angular momentum

Angular momentum is a vector quantity. This quantity is often termed as the moment of inertia. In which a rotational body shows movement concerning inertia. Also, this momentum acts as a resisting force against the angular acceleration change. Moreover, the law of conservation of Angular momentum depicts that any object’s physical property of angular momentum does not change with time if in a closed system. 

This law was formulated alongside two other conservation laws. All these laws were prepared with the help of Sir Isaac Newton’s motion laws and Newton did not directly acquire them and science discovered them later. 

Law of conservation of angular momentum statement

The Law of conservation of angular momentum states that if no external torque continues to act on an object, no change in angular momentum will take place. Moreover, in simpler words, this Law is an analogue of linear momentum where the angular momentum exchange takes place. Yet, the total angular momentum before the exchange is the same as the post-exchange.    

Law of conservation of angular momentum formula

The total angular momentum of objects or a body is often denoted as L. Moreover, Law of conservation of angular momentum formula can be represented as. (t = time)

dLdt=0

 Or, 

L=L1+ L2+L3++Ln=Constant

The Law of conservation of Angular momentum explains that individual angular moments can happen as long as their total amount remains unchanged. Though, whenever the external factor on a process is zero, this law shows conservation of linear momentum. 

Example of the law of conservation of angular momentum

• When a springboard swimmer jumps into a pool, he adjusts himself in a somewhat way that his top and lower legs are forced close together.

• Ballet dancers stretch their legs towards their hands during dancing. 

• Our solar system’s celestial bodies are in elongated circular orbits, with the sun at one of its focuses.

• The inner layers of wind move faster than the outer layers.

• The loss in kinetic energy of a flywheel system when coupled with another wheel.

Illustration

A gearwheel spins without friction at such an angular speed of 36000 rev per hour on a metal rod with zero friction metal rod with minimal rotational inertia. Another stationary flywheel with an angular momentum three times those of the spinning flywheel is gone down onto it. But since friction arises in both the planes, the flywheels easily reach the same angular velocity and spin together.  Now, using the Law of Conservation of Angular Momentum, determine the angular speed of the combination.

Solution: 

Let the first wheel be ‘IΘ’, therefore, second wheel will be, ‘3IΘ’. The system is not subjected to any external torques. The frictional force generates an internal torque that has no effect on the system’s angular momentum. As a result, angular momentum conservation provides

IΘ . ωΘ = (IΘ + 3I0) ω, 

ω = ¼ ωΘ  ; (ωΘ = 36000 revolution per hour = 600 revolutions per min)

ω = ¼ 600 revolutions / min 

= 150 revolutions / min 

= 942.48 radian / min (1 rad = 6.28rev)

= 15.7 radian / sec.

Conclusion

We came through various aspects of the law of conservation of angular momentum from all the above. We learned that angular momentum is the vector quantity of any object and the Law of conservation of momentum statement says an angular momentum remains constant against the exchange. Moreover, we also discussed simplifying examples and numerical for easy understanding.